Tensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in $(2+1)$ dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages—firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse-graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentanglers. We apply this new algorithm to lattice gauge theories defined for the quantum group $\mathrm{SU}{\left(2\right)}_{\mathrm{k}}$ and identify a weak and a strong coupling phase for various levels $\mathrm{k}$ . As we increase the level $\mathrm{k}$ , the critical coupling $g_c$ decreases linearly, suggesting the absence of a deconfining phase for the continuous group $\mathrm{SU}\left(2\right)$ . Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases.

中文翻译：

---

具有融合电荷的张量网络重归一化—在3D格规理论中的应用

张量网络方法是研究统计和量子系统的特性和动力学的强大而有效的工具，特别是在一维和二维中。近年来，这些方法已应用于晶格规理论，但是这些理论仍然是挑战。 $（2+\mathrm{1个}）$ 尺寸。在本文中，我们提出了一种新的（装饰的）张量网络算法，其中，张量编码以融合为基础表示的晶格规幅。这具有几个优点-首先，融合基础确实使操作员对角线化，从而测量与一组分层区域关联的磁通量和电荷。因此，该算法允许直接访问这些可观察对象。其次，与先前使用的自旋网络基础相反，融合基础在粗粒度下是稳定的。第三，由于融合基础的层次结构，该算法确实实现了预定义的解缠结器。我们将此新算法应用于为量子群定义的晶格规理论 $苏{（2）}_{\mathrm{ķ}}$ 并确定各个级别的弱耦合阶段和强耦合阶段 $\mathrm{ķ}$ 。随着我们提高水平 $\mathrm{ķ}$ ，关键耦合 $G_C$ 线性下降，表明连续组没有解围阶段 $苏（2）$ 。此外，我们说明了两个阶段中Wilson循环的缩放行为。